Seohee So^{1}, Jaejin Cho^{1}, Kinam Kwon^{1}, Byungjai Kim^{1}, Wonil Lee^{1}, and Hyunwook Park^{1}

Magnetic resonance thermometry provides noninvasive temperature measurements for thermal therapy. In this abstract, we exploit linear phase relation generated by echo shifting in the bSSFP acquisition to measure PRF change. Echo-shifting from TE=TR/2 in bSSFP provides a linear relation between phase of transverse magnetization and phase evolution in TR. This linearity enables frequency prediction from the phase information, which makes temperature measurement with PRF shift possible. The performed simulations show shifted-echo bSSFP of TE=TR/4 well estimates frequency change.

The MR signal from the balanced SSFP is usually acquired at the half of repetition time, i.e., TE=TR/2. When TE=TR/2, there is no linear relation between phase of transverse magnetization and off-resonance frequency as shown in figure 1(b). In the simulation, we use TR=10ms, flip angle=60°, and T1/T2=3800/1400ms for water. If TE is shifted from the half of the TR, linear phase with respect to off-resonance frequency is generated. Magnitude and phase of bSSFP signal are formulated as follows :

$$\left| M_{xy} \right| = \left| \frac{M_{0}\left(l-E_{1}\right) \sin\alpha E_{2}^{'}}{\left(1-E_{1}\cos\alpha\right)\left(1-E_{2}\cos\left(-\theta+\psi \right)\right) -E_{2}\left(E_{1}-\cos\alpha\right)\left(E_{2}-\cos\left(-\theta+\psi \right)\right)} \right|\sqrt{\left(1-E_{2}\cos\left(-\theta+\psi \right)\right)^{2}+E_{2}^{2}\sin^{2}\left(-\theta+\psi \right)}$$

$$\angle M_{xy} = \tan^{-1} \frac{-\sin\delta -E_{2}\sin\left(-\theta+\psi+\delta \right)}{-\cos\delta -E_{2}\cos\left(-\theta+\psi+\delta \right)} \approx -\theta\left(\frac{1}{2}-\frac{TE}{TR}\right)+\frac{\psi}{2}-\frac{\pi}{2}\left(2\times\lfloor \frac{-\theta+\psi}{2\pi} \rfloor +1\right) ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ \tt eq.(1)$$

where $$$M_{0}$$$ is the net magnetization, α is a flip angle, ψ is a phase cycling angle, $$$E_{1}$$$ is $$$e^{-TR/T_{1}}$$$, $$$E_{2}$$$ is $$$e^{-TR/T_{2}}$$$, $$$E_{2}^{'}$$$ is $$$e^{-TR/T_{1}}$$$, θ represents phase evolution in TR, and δ represents phase evolution in TE after excitation. Phase of transverse magnetization could be approximately proportional to θ with assumption $$$E_{2} \approx1$$$. This assumption is quite reasonable when –θ+ψ is a multiple of 2π and $$$T_{2}$$$ is long as shown in figure 1(c). For the shifted echo of $$$TE \neq TR/2$$$, the phase of transverse magnetization is proportional to θ. At TE=TR/4, phases and θ have one-to-one correspondence with maximum slope in 4 cycles corresponding to 8π as shown in figure 1(d). θ could be estimated from phase information easily and it is shown in figure 1(e). Nonlinearity near banding points induces prediction error. To correct this error, one more bSSFP acquisition whose phase cycling angle is different from the first acquisition could be used. We use phase cycling angle 0 and π radian. Near banding region, signal magnitude drops drastically. Comparing magnitude of the two acquisitions, the phase is selectively assigned as follows:

$$\left(phase, \theta \right) = \begin{cases}\theta_{PC 0} & |M_{xy\_PC0}|>\frac{3}{2}\times|M_{xy\_PC\pi}| \\\theta_{PC \pi} & |M_{xy\_PC\pi}|>\frac{3}{2}\times|M_{xy\_PC0}| \\ \frac{1}{2}\times(\theta_{PC 0}+\theta_{PC \pi}) & otherwise \end{cases} ~~~~~~~~~~~~~~~~~~~~ \tt eq.(2)$$

Combined phase appropriately compensates prediction error as shown in figure 2.

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(a) bSSFP
pulse sequence, (b),(c) magnitude and phase at different TE with TR=10ms, (d)
phase of transverse magnetization and (e) reconstructed phase evolution per TR
using the phase of transverse magnetization.

(a)
Magnitude and (b) phase of transverse magnetization with phase cycling angle 0
and π radian. The yellow and purple dots are selected by eq.(2). (c) The
reconstructed phase evolution per TR with phase cycling angle 0 and π radian,
and the combined phase.

Phantom simulations. (a) Magnitude and (b)
estimated phase evolution in TR with phase cycling 0 and π radians before
heating, (c) selectivity map and combined phase before heating. (d)-(f)
Magnitude and phase maps corresponding to (a)-(c), respectively, after heating.
(g) True frequency change due to heating. (h)-(i) Estimated change of frequency
and cutview (down) corresponding to PC 0, PC π, respectively and (j) the
combined result.